Fl Begiltlteri Fiuide Tu
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flBegiltlteri fiuidetu Iullrtru rtln g Ifl |jll lve rre Iheffiuthemuticul flrchetgpBrnfllnttlre, frt,rlld Icience [|lrttnrrI ftttnn[tR HARPTFI NEW Y ORK LO NDON TORONTO SYDNEY Tomy parents, SaIIyand Leonard, for theirendless loae, guidance,snd encouragement. A universal beauty showed its face; T'heinvisible deep-fraught significances, F1eresheltered behind form's insensible screen, t]ncovered to him their deathless harmony And the key to the wonder-book of common things. In their uniting law stood up revealed T'hemultiple measures of the uplifting force, T'helines of tl'reWorld-Geometer's technique, J'he enchantments that uphold the cosmic rveb And the magic underlving simple shapes. -Sri AurobitrdoGhose (7872-1950, Irrdinnspiritunl {uide, pttet ) Number is the within of all things. -Attributctl to Ptlfltngorus(c. 580-500s.c., Gre ek Tililosopther nrr d rtuttI rc nut t ic inn ) f'he earth is rude, silent, incomprehensibie at first, nature is incomprehensible at first, Be not discouraged, keep on, there are dir.ine things well errvelop'd, I swear to vou there are dirrine beings more beautiful than words can tell. -Wnlt Whitmm fl819-1892,Americnn noet) Flducation is the irrstruction of the intellect in the laws of Nature, under which name I include not merely things and their forces,but men and their ways; and the fashioning of the affections and of the r,t'ill into an earnest and loving desire to move in harmony with those laws. -Thomas Henry Huxley (1 B 25-1 Bg 5, Ett glislth iolo gis t) Contents ACKNOWLEDGMENTS xi GEOMETRY AND THE QUEST FOR REALITY BY JOHN MICHELL xiii INTRODUCTION xvii MONAD WHOLLY ONE 2 DYAD IT TAKES TWO TO TANGO 21 3 TRIAD THREE-PART HARMONY 38 4 TETRAD MOTHER SUBSTANCE 60 5 PENTAD REGENERATION 96 6 HEXAD STRUCTURE-FUNCTION-ORDER 178 7 HEPTAD ENCHANTING VIRGIN 221 8 OCTAD PERIODIC RENEWAL 267 9 ENNE AD THE HORIZON 301 10 DECAD BEYOND NUMBER 323 EPILOGUE NOW THAT YOU'VE CONSTRUCTED THE UNIVERSE ... 347 ix fienmetryand thefluestfurfleality lohn Michell Sooner or later there comes a time in Me when you start thinking about Reality and where to find it. Some people tel l you there is no such thing, that the world has nothing per- manent in it, and, as far as vou are concemed, consists merely of your fleeting experiences.Its framework, they say, is the random product of a nafural process,meaningless and undirected. Others believe that the world was made by a divine Cre- ator,who continuesto guide its development.This soundsa more interesting idea, but, as skeptics point out, every reli- gion and church that upholds it does so by faith alone. If you are naturally faithful and can accept without question the orthodoxy of your particular religion or system of beliefs, you will feel no need to inqufue further and this book will appear superfluous. It was written for those of us who lack or have lost the gift of simple faith, who need evidence for our beliefs. We cannot help being athacted by the religious view, that the world is a harmonious, divinely ordered cre- ation in whidr, as Plato prornised the uninitiated, "things are taken care of far better than you could possibly believe." Yet superficially it is a place of confusion and chaos, where suffering is constant and the ungodly flourish. ANDTHE OUEST FOR REALIry xlv OEO}IETRY This is where we begin the quest for Reality' Lgokinq closelv at nature, the firsi insight we obtain is thai, behind the apparently endlessproli{eration of natural objects,there is a flr lesser number of apparently fixed tlpes' We see' for example,that through every generationcats are cats and are the same way, every 'ro"""hu"progrlmmed for catlike behavior. In the unique characteristics of a rosb and every oak these leaf is definitely an oak leaf. No two specimens of -are ever exactlythe same,but eachone is clearly a product of its formative i1pe. If it were not so, if animals and plants,sim- ply inherited their progenitors' characteristics,the order of nature would soon dissolve into an infinite variety of crea- tures, undifferentiated by speciesand kinships' This observation, of one type with innumerable prod- ucts, gives rise to the old philosophical problem o{ the One and the Manv. The problem is that, whereas the Many are visible and tangible ind can be examined at leisure, the One is never seen or sensed, and its very existence is only inferred tfuough the evident effect it has upon its products, the Many. Yet, paradoxically, the One is more truly real than the Many. In the visibte world of nature all is flux' Every- thing is either being bom or dying or moving between the two processes.Nothing ever achievesthe goal of perfection o. ti'," ttute of equilibrium that would allow it to be describedin essencb.The phenomenaof nature, said Plato, are always "becorning," never actually " are." Our fiue sensestell us that they are real, but the iniellect judges dif- ferently, reasoning that the One, which is constant, creative, and ever the sarne, is more entitled to be called real than its ever-fluctuating products. The search for Reality leads us inevitably toward the type, the enigmatic One that lies behind the obvious world of*re Many. Imrnediately we encounter difficulties. Being imperceptible and existing only as abstractions, tyPes can- not be apprehendedby the methods of physical science.A number of modem scientists, perceiving the influence of types in nature, have attempted to bring them within the range of empirical study. Rupert Sheldrake, author of A Nezu Scienceof Life and other works, has taken a bold step in that direction.In an earlier age,the Pythagoreansworked on sys- tematizing the types by means of numerical formulae. Yet SEOiETRY AND TH€ OUESTFOR REALITY Plato, who wrote at length on the subject of constant types (referred to as "forms"), was carefully ambiguous in defin- ing them and never made clear the means by which they influence the world of appearances. Plato did, however, give instructions on the procedue toward understanding the nature and function of the types. In the Republiche described the ascent of the mind through four different stages.It begins in Ignorance, when it does not even know that there is anything worth knowing. The next stage is Opinion, the stage in which TV chat-show partici- Dants are forever stuck. This is divided into two subcate- gories, Right Opinion and Wrong Opinion. Above that is the Ievel of Reason.By education and study, particularly in cer- tain mind-sharpening subjects,the candidate is prepared for entry into the fourth stage, which is called Intelligence (nous). One can be prepared for it but with no guaranteeof success,for it is a level that one can only achieve on one's own, the level of heightened or true understanding, which is the mental level of an initiate. The studies that Plato specified as most effective in preparing the rnind for understanding are the so-calied mathematical subjects, consisting of number itself, music, geometry, and astronomy. These were the main studies of Plthagoras and his followers, who anticipated the realiza- tion of modem physics in proclaiming that all scales and departrnents of nature were linked by the same code of number. Geometry is the purest visible expression of num- ber. In Platonic terms, the effect of its study is to lead the mind upward from Opinion onto the level of Reason,where its premises are rooted. It then provides the bridge or ladder by which the mind can achieve its highest level in the realm of pure Intelligence. Geometry is aiso the bridge between the One and the Many. When you draw one of its basic figures-a circle, say, or a triangle or regular polygon-you do not copy someone else's drawing; your model is the abstract ideal of a circle or triangle. It is the perfect form, the unchanging, unmanifest One. Below it are the Many-the expressions of that figure in design, art, and architecture. ln nature also the One circle gives rise to the Many, in the shapes and orbits of the plan- ets, in the roundness of berries, nests, eyeballs, and the xvi GEO}IETRYAND THE OUEST FOR REALITY cycles, of time. On every scale, every nafural pattern of growth or movement conforms inevitably to one or more of the simple geomerrictypes. The pentagoryfor example,Iies behind each specimen of the five-pelaled rose, the five_ fingered starfish, and many other living forms, whereas the sixfold, hexagonal gpe, as seen in the structure of snowflakes and crystal generally, pertains typically to inan_ imate nature. As soonas you enter upon the world of sacred,symbolic, or phiJosophicaigeometry-from your first, thoughifuJcon_ struchon of a circle with the circumferencedivid;d into its natural six parts-your mind is opened to new influences that stirnulate and refine it. you begin to see,as never before, the wonderfuily patlerned beauty-of Creation. you see true artistry, {ar above any human contrivance. T?risindeed is the very, source of art. By contact with it your aestheticsenses arc heightenedand set_uponthe firm bisis of truth. Beyond the oovlous pteasureot contemplatingthe works of nafure_the MTy-1" the delight that comes through the philosophical study of geometry, of moving toward-the prr".,." of th" One. Michael Schneideris an experiencedteacher and, as you areentitled to expect,a masterof his geometriccraft. No Lne less qualified could set out its basic piinciples so ctearty anJ simply. His much rarer assetis appieciation of the symbolic and,cosmologicals].'rnbolism injrerent in geometryiThat is the best reasonf9r being interestedin the subject,and it is why.